My violin bow vibrates when I play down bow, especially on e string
Jared, if you think " my bow arm vibrates with down bow" you got it halfway solved.
If you mean it bounces on a bold/loud down-stroke, then try starting the stroke closer to the frog. You'll also need to make sure the pressure you apply is steady.
If you're talking about the bow vibrating in the middle of the bow stroke you need to be sure you've got a proper grip on the bow and maintain constant slight pressure with your index finger to counter the tendency of the bow to bounce as it travels across the string. Keep your arm relaxed and in proper posture but I have found that when my bow bounces I'm not maintaining proper pressure on the string. Not heavy pressure, just consistent pressure.
Vibrating, or bouncing / trembling?
Jared, It's time to explain more fully the phenomenon troubling you.
You might be interested in Nathan Cole's youtube video, "Safety move eliminates violin bow shakes." As he says, the bow is made to bounce. I have the most trouble on down bows in the lower half of the bow. What helps me the most is to make sure my bow is tracking parallel to the bridge. And, some bows seem to shake more than others. But apparently all do to some degree.
Sometimes people shake, too.
Previous responders appear to have assumed your experience is driven by your hand. If it is not, try this:
Andrew, thank you for the link to Gough's paper on bow vibrations. It's good to see some serious math being put to work in explaining a phenomenon we've all experienced some time or other. It's what one would expect - the vibrations of a string interacting with the natural vibrations of a moving bow that is causing the string to vibrate.
If it's a bow phenomenon rather than muscular, it's related to camber and a REALLY GOOD bow maker can remove the vibration. I've seen this done many times, with spectacular success by people who know exactly how it's done.
So now, like often, the person posting a question is getting a lot of good answers but doesn't care to reply or react in any way.
Sometimes it is annoying when the OP doesn't make any further comment (especially if the question is contentious), but my finding is that quite often if I interfere with answers to a question I ask, all I do is muddy the waters, so it's sometimes better to see how the conversation goes under its own steam, rather than try to direct it. e.g. I might try to refine a definition only to find people think I'm redefining the entire question (indeed I am, if they misunderstood in the first place, but by then the whole thing is usually shot to hell anyway).
As Michael said, there is the possibility of an issue with the camber. A bow can tremble in certain sections when there is a flat spot in the curve. Take it to a bow maker to have a look if the problem still persists.